Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Solve each triangle. \(C=28.3^{\circ}, b=5.71\) inches, \(a=4.21\) inches

Step-by-Step Solution

Answer

The triangle is solvable with angles B ≈ 42.67°, A ≈ 109.03°, and side c ≈ 2.145 inches.

Step 1: Determine Triangle Type

The given values imply this is an ambiguous case (SSA, side-side-angle). We need to first identify the number of possible solutions by determining if a valid triangle can be formed. This step involves comparing the given angles and side lengths.

Step 2: Apply Law of Sines

Use the Law of Sines to find angle B. The formula is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Since angle C is known, use \( \frac{a}{\sin A} = \frac{b}{\sin B} \) thus, \( \sin B = \frac{b \cdot \sin C}{a} \). Substitute the known values to find \( \sin B = \frac{5.71 \cdot \sin(28.3^{\circ})}{4.21} = 0.6767 \).

Step 3: Find Angle B Using Inverse Sine

Calculate angle B using the inverse sine function. \( B = \sin^{-1}(0.6767) \approx 42.67^{\circ} \).

Step 4: Find Angle A

Now calculate angle A using the fact that the sum of angles in a triangle is 180°. \( A = 180^{\circ} - B - C = 180^{\circ} - 42.67^{\circ} - 28.3^{\circ} = 109.03^{\circ} \).

Step 5: Calculate Side c Using Law of Sines Again

Utilize the Law of Sines to find the length of side c. \( \frac{c}{\sin C} = \frac{a}{\sin A} \) which simplifies to \( c = a \cdot \frac{\sin C}{\sin A} = 4.21 \cdot \frac{\sin(28.3^{\circ})}{\sin(109.03^{\circ})} \approx 2.145 \) inches.

Key Concepts

Law of Sines

The Law of Sines is a powerful tool when solving triangles, especially when we have certain angles and side lengths given. It relates the lengths of the sides of a triangle to the sines of the angles opposite those sides. The formula is expressed as: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.\]This law applies to any kind of triangle, not just right triangles. It is particularly useful in the SSA (Side-Side-Angle) triangle case. To find an unknown side or angle, choose the known ratio and solve for the unknown variable. This often involves rearranging the equation to isolate the sine of the desired angle. This law is fundamental in techniques for solving triangles and is often the first step when multiple angles or sides are known.

• Choose ratios with known values.
• Use a calculator for trigonometric functions.
• Ensure your calculator is in degree mode (if using degrees).

Practicing with the Law of Sines increases efficiency when dealing with different types of triangles.

SSA Triangle

An SSA triangle scenario involves knowing two sides and a non-included angle, noted as Side-Side-Angle. This scenario can be tricky because it can result in zero, one, or two solutions. This is known as the ambiguous case.In our exercise, given \( b = 5.71\) inches, \( a = 4.21\) inches, and \( C = 28.3^{\circ} \), we're dealing with an SSA configuration. Calculations must check for the possibility of having two different triangles.

• Compare the given side "opposite" the angle with the "base" side.
• Check if there’s a need for evaluating two possible angles using the inverse function.
• Confirm the validity of any found angles with the angle sum property.

This can sometimes lead to unexpected results, where calculations might suggest a triangle is not possible or that two different triangles might fulfill the conditions.

Inverse Sine Function

The inverse sine function, often denoted as \( \sin^{-1} \) or arcsin, is used to find an angle when its sine is known. This is crucial in solving triangles, especially in the SSA scenario. For an angle \( B \) with known \( \sin B \), the calculation is straightforward: \[B = \sin^{-1}(\text{value})\]In our exercise, after finding \( \sin B = 0.6767 \), we used the inverse sine to calculate \( B \approx 42.67^{\circ} \).

• The inverse sine function gives the angle in the range \(-90^{\circ} \leq \theta \leq 90^{\circ}\).
• Check for secondary solutions (e.g., supplementary angles, since \( \sin(\theta) = \sin(180^{\circ} - \theta) \)).
• Use a calculator and double-check the mode setting (degrees versus radians).

Remember to account for possible alternative angles in the SSA scenario, which might require additional steps.

Angle Sum Property

The angle sum property is an essential part of triangle calculations. It states that the sum of the interior angles in any triangle is always \(180^{\circ}\). This rule helps to find the remaining angle(s) when two angles of a triangle are known. In your exercise, after finding angles \( B \) and \( C \), the remaining angle \( A \) is calculated as:\[A = 180^{\circ} - B - C\]This gives a quick and reliable way to ensure that all angles in a triangle sum up correctly.

• Always check if the computed angles satisfy the angle sum property.
• This property confirms the validity of your calculated angles, especially in ambiguous cases.
• Use this property to quickly find the last angle once two angles are known.

Mastering this simple yet powerful concept ensures that working with triangle geometry becomes more intuitive.